Bouncing models in an extended gravity theory

Observations from high redshift supernova, Cosmic Microwave Background (CMB) radiation anisotropy, large scale structure and Baryon Acoustic Oscillation have confirmed that the universe is accelerating at the present epoch. This late time cosmic speed up issue has led to the development of many new ideas and concept. In the framework of General Relativity(GR), this phenomenon has been attributed to an exotic energy form known as dark energy (DE). In its usual form, GR is not able to explain this bizarre fact. Therefore, additional dynamical degrees of freedom mediating through scalar fields are required Zlatev1999 ; Steinhardt1999 . In GR, this DE corresponds to a mysterious fluid with negative pressure. Simplest possible DE candidate is the cosmological constant Sahni2008 ; Weinberg1989 . However, the cosmological constant is plagued by many fundamental issues such as the fine tunning problem and coincidence problem. In order to avoid the cosmological constant problem, slowly rolling scalar fields like quintessence have been proposed Copeland2006 ; Caldwell2009 . Most of the dynamical scalar fields are ghost fields with unusual negative kinetic energy at least around flat, cosmological or spherically symmetric backgrounds, e.g. Bulware-Desser mode in massive gravity Bulware1972 , bending mode in the self-accelerating branch of Dvali-Gabadadze-Porrati model Koyama2007 ; Sbisa2015 ; Gumrukcuoglu2016 .

Geometrically modified theories of gravity have been proposed in recent times as alternatives to GR, that do not require additional degrees of freedom to explain the late time cosmic acceleration and therefore provide ghost free models. In this alternative approach, the Einstein-Hilbert action is modified by considering a more general function of $R$ or by a matter-geometry coupled functional $f(R,T)$ in place of the Ricci scalar $R$. There are so many geometrically modified theories of gravity have been proposed in literature. Notable among them are $f(R)$ theory Caroll2004 ; Nojiri2007 ; Bertolami2007 , $f(G)$ gravity Nojiri2005 ; Li2007 , $f(\mathcal{T})$ theory Linder2010 ; Myrzakulov2011 and $f(R,T)$ theory Harko2011 . Among these modified theories, the $f(R,T)$ gravity theory Harko2011 has attracted a lot of research attention in recent times and has been widely used to address many issues in cosmology and astrophysics Zubair2015 ; Mishra2016 ; Yousaf2016 ; Singh2018 ; Tretyakov2018 ; Velten2017 ; Wu2018 ; Baffou2017 ; Carvalho2017 ; Mishra2018a ; Baffou2019 ; Houndjo2014 ; Das2017 ; Deb2018 ; Biswas2020 . Extended gravity theories are a specific extension of the $f(R,T)$ gravity theory where the functional $f(R,T)$ contains a term linear in $R$. Recently, there has been an increased interest in extended theories of gravity because of its simple structure and elegance in addressing issues in cosmology Capozziello2011 ; Mishra2018b ; Mishra2018c ; Capozziello2019 ; Tripathy2020 .

The cosmological evolution in the early Universe is usually described by the Standard Big Bang cosmology. However, the Standard cosmology model suffers from many issues such as the flatness problem, the horizon problem, initial singularity and baryon asymmetry problem. The flatness problem and the Horizon problem are concerned with the questions: why is the density of universe so close to the critical density? and why does the universe (or CMB radiation) look the same in all directions? The inflationary scenario solved some of these issues of the early Universe standard model and provided a causal theory of structure formation Guth1981 ; Mukhanov1981 . In spite of great success, the inflationary scenario suffers from the singularity problem and the trans-Planckian problem for fluctuations. At the onset of inflation, the Universe undergoes an almost exponential expansion. Before such an inflationary phase, as an obvious phenomenon singularity occurs. Therefore inflationary scenario fails to reconstruct the complete past history of the Universe.

An alternative model to the Standard Big Bang scenario, describing inflation without initial singularity had been proposed which is known as the emergent universe Ellis2004 . Another non-singular approach without inflation known as the matter bounce scenario have been proposed Bars2011 ; Bars2012 ; Brandenberger2011 ; Brandenberger2012 . For some reviews see Novello2008 ; Brandenberger2016 ; Battefeld2014 .

The initial singularity occurring in the Standard Big Bang cosmology and the inflationary cosmology can be suitably avoided in the matter bounce scenario. In bouncing scenario, the Universe undergoes an initial matter dominated contraction phase followed by a non-singular bounce and then there is a causal generation for fluctuation Novello2008 ; Brandenberger2016 . Moreover, in the contracting phase the scale factor decreases ($\dot{a}<0$) and in the expanding phase, scale factor increases ($\dot{a}>0$) and at the matter bounce epoch, we have $\dot{a}=0$. Consequently, in bouncing cosmology, the Hubble parameter $H$ passes across zero ($H=0$) from negative values $H<0$ to positive values $H>0$ Tripathy2019 ; Tripathy2020a . Bouncing cosmologies have been investigated in alternative gravity theories such as $f(R)$ theory Bamba2014a ; Barragan2009 ; Barragan2010 ; Saidov2010 ; Chakraborty2018 , modified Gauss-Bonnet gravity Bamba2014b ; Bamba2015 , $f(R,T)$ gravity Singh2018 ; Tripathy2019 ; Sahoo2020a and $f(T)$ gravity Cai2011 . Resolving the initial singularity problem applying Loop Quantum Gravity approach gives rise to a cosmological scenario which can be considered to be a combination of the emergent and bouncing universe picture Alesci2017 .

In the present work, we have investigated some bouncing models in the framework of an extended gravity theory. Our motivation is to use a simple extended gravity theory exploring the geometrical degrees of freedom to explain the late time cosmic speed up phenomenon and to investigate the bouncing behavior at an initial epoch. We have considered some widely known bouncing models described through specific scale factors in the framework of extended gravity and studied the cosmic dynamics. The paper is organized as follows: in Sec. II, the basic formalism of the extended gravity theory and the field equations for a flat FRW universe have been derived. In Sec. III, dynamical physical parameters such as the energy conditions and the equation of state (EOS) parameter are derived in terms of the Hubble parameter. Three different bouncing models along with their evolutionary behavior have been studied in Sec. IV. At the end the conclusion and summary are presented in Sec. V.

The action for a geometrically modified extended theory with a matter-geometry coupling is considered as

where $\mathcal{L}_{m}$ is the matter Lagrangian. $f(T)$ is an arbitrary function of the trace $T$ of the energy-momentum tensor. The unit system followed in writing the action is the natural unit system where $8\pi G=c=1$; $G$ and $c$ being the Newtonian gravitational constant and speed of light in vacuum respectively. The interesting fact of the above action is that it reduces to that of GR for a vanishing $\beta$.

Varying this action with respect to the metric $g_{\mu\nu}$, the modified field equations are obtained as

where the energy-momentum tensor $T_{\mu\nu}$ is defined through the matter Lagrangian as

and $\Theta_{\mu\nu}=g^{\rho\sigma}\frac{\delta T_{\rho\sigma}}{\delta g^{\mu\nu}}$. Here $f_{T}(T)$ denotes the partial derivative of $f(T)$ with respect to $T$.

We chose a perfect fluid distribution of the universe with energy density $\rho$ and pressure $p$ for which the energy-momentum tensor is given by

where $u_{\mu}$ is the time like four-velocity vector of the cosmic fluid that satisfies the relation $u_{\mu}u^{\mu}=1$ in co-moving coordinates. In this context we also chose the matter Lagrangian as $\mathcal{L}_{m}=-p$, so that $\Theta_{\mu\nu}=-pg_{\mu\nu}-2T_{\mu\nu}$.

The modified field equation now reduces to

Simple algebraic manipulation leads to

where $G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}$ is the usual Einstein tensor and $\kappa_{T}=1+\beta f_{T}(T)$ is the redefined Einstein constant. The redefined Einstein constant may be a dynamical quantity for a non linear choice of the functional $f_{T}(T)$. However, for a linear choice of the functional $f_{T}(T)$ it becomes a constant carrying the signature of the extended gravity through the coupling parameter $\beta$. In Eq. (6), we have

The geometrical modification in the action leads to the extra effective interaction term $T^{int}_{\mu\nu}$ which is necessary to provide the required acceleration at late times. It is worthy to mention here that, such a coupling of matter and curvature is motivated from quantum effects and particle creation process. In the limit $\beta\rightarrow 0$, the interaction term vanishes. In principle one may construct viable cosmological models through suitable choices of the functional $f_{2}(T)$ which may be confronted with recent observations. In the present work, we consider a linear functional

so that

and

The field equations in the extended gravity theory for a flat FRW model become

Here $\alpha=1+3\beta$ and we denote the ordinary time derivatives as overhead dots. The Hubble parameter is defined through the scale factor $a(t)$ as $H=\frac{\dot{a}}{a}$.

The physical parameters of the models such as the pressure and energy density can be obtained from the field equations (11) and (12) in terms of the Hubble parameter as

The energy condition $\rho+p\geq 0$ and $\rho\geq 0$ put extra constraints on the model. The positive energy density constrains the coupling parameter through the condition $\frac{3\kappa_{T}}{2\beta}\geq\frac{\dot{H}}{H^{2}}$.

The equation of state (EOS) parameter can be obtained as

The EOS parameter can also be expressed in terms of the deceleration parameter $q=-1-\frac{\dot{H}}{H^{2}}$ as

This is an important equation in the sense that, the time dependence of the deceleration parameter controls the time evolution of the equation of state parameter. A constant EOS will result from a constant deceleration parameter.

The null energy condition (NEC) is expressed as $\rho+p\geq 0$. In the purview of the extended gravity theory, from Eqs. 13 and 14, we can have

Since $\kappa_{T}=1+2\beta$ is a positive quantity for a small value of $\beta$, the NEC is violated for positive values of the Hubble parameter. In fact, in phantom models, the NEC is usually violated.

In this section we have considered three different scale factors that show bouncing behavior at some epochs. The bouncing nature of these scale factors are discussed. Also, we derive the dynamical properties such as the deceleration parameter and the EOS parameter for these models.

The stability of the three bouncing models can be analyzed under linear homogeneous perturbations in the FRW background. We consider linear perturbations for the Hubble parameter and the energy density as Sharif2013 ; Sahoo2020b

where $\delta(t)$ and $\delta_{m}(t)$ are the perturbation parameters. In the above, we have assumed a general solution $H(t)=H_{b}(t)$ which satisfies the background FRW equations. The matter energy density can be expressed in terms of $H_{b}$ as

In the extended gravity theory with the functional $f(R,T)=R+2\beta T$, the Friedman equation and the trace equation are obtained as

where $\mathbf{\Theta}=3H$ is the expansion scalar. The first order perturbation equation for a standard matter is given by

Using Eqs. (46) - (47), we obtain

An elimination of $\delta(t)$ from Eqs. (51) and (52) yields the first order matter perturbation equation

which on integration leads to

where $C$ is a non zero positive constant. The evolution of the linear perturbation $\delta(t)$ reads as

Near the bouncing epoch, we have $H_{b}(t)=0$ which blows up the factor $\frac{T}{H_{b}}$ in the integrand of Eqs. (54) and (55). Therefore the model is unstable near the bouncing epoch. However, all the bouncing models considered in the present work, the integral $\int\frac{T}{H_{b}}~dt$ are evaluated for positive time domain which comes out to be positive. Consequently, the linear perturbation $\delta(t)$ smoothly decay out to provide stability to the models.

In the present work, we have discussed some bouncing cosmological models in the frame work of a simple extended theory of gravity. The extended theory of gravity is derived from an action where the usual Ricci scalar is replaced by a minimally coupled function linear in $R$ and $T$. The presence of the trace of the energy momentum tensor in the geometry side of the Einstein-Hilbert action leads to an additional interactive term in the field equations. This matter field dependent additional interactive term is responsible for a late time cosmic acceleration as predicted from a lot of observational data. Such a concept to include a bit of matter field within the gravitational action is motivated from quantum effects and particle production process and can be associated with the existence of imperfect fluids. This extended gravity theory is simple to handle mathematically but can be elegant in explaining many issues in cosmology and astrophysics.

In the frame work of such an extended gravity theory, we have investigated three different models that show bouncing behavior at some epochs. All the three models explain the late time acceleration. Model II in the limit $\lambda\ll 1$ similar as Model I for $\lambda_{1}=\lambda_{2}=\lambda$. However, the deceleration parameter shows contrasting behavior for Model I and Model II although the models share some similarities. In the first two models the bounce takes place at $t_{b}=0$ but for Model III the bounce takes place at a finite time $t_{0}$. There might be finite time singularities present in Model III for non-integral positive values of $n$ due to the occurrence of a saddle point in the scale factor at $t=t_{0}$. In these models we have shown the effect of the coupling parameter of the extended gravity theory in lifting the omega-singularity occurring in GR. The presence of a finite non-zero but small values of $\lambda$ removes the singularity occurring in the equation of state parameter during the bouncing epoch.

It is important to note that, the evolutionary behavior of the EOS parameter is mostly decided by the coupling parameter of the extended gravity. We have derived the energy conditions for the models and observed that for all the models, the energy conditions are violated. In fact, a violation of energy condition is an integral part in the bouncing scenario. Also, the violation of energy conditions enable the model with a positive slope for the Hubble parameter to evolve in the phantom phase. We have analyzed the stability of the bouncing models under a linear homogeneous perturbations in the FRW background. At the bouncing epoch, the models are highly unstable but as we move away from the bounce in the positive time domain, the perturbation smoothly decay out ensuring stability of the models at late times.

S.K.T., B.M., and S.R. are thankful to the Inter-University Centre for Astronomy and Astrophysics, Pune, India for providing the Visiting Associateship under which a part of this work was carried out.